Collaborative robot manifold tracker

ABSTRACT

A collaborative control method for tracking Lagrangian coherent structures (LCSs) and manifolds on flows employs at least three autonomous sensors each equipped with a local flow sensor for sensing flow in a designated fluid medium, e.g. water or air. A first flow sensor is a tracking sensor while the other sensors are herding sensors for controlling and determining the actions of the tracking sensor. The tracking sensor is positioned with respect to the herding sensors in the fluid medium such that the herding sensors maintain a straddle formation across a boundary; obtaining a local fluid flow velocity measurement from each sensor. A global fluid flow structure is predicted based on the local flow velocity measurements. In a water medium, mobile autonomous underwater flow sensors may be deployed with each tethered to a watersurface craft.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application61/638,694 filed on Apr. 26, 2012, and incorporated herein by reference.

FIELD OF THE INVENTION

The invention is directed to tracking coherent structures and manifoldson flows in a designated fluid medium, and in particular to deployingmobile autonomous underwater flow sensors, e.g. each tethered to awatersurface craft, to track stable/unstable manifolds of general 2Dconservative flows through local sensing alone.

BACKGROUND OF THE INVENTION

In realistic ocean flows, time-dependent coherent structures, orLagrangian coherent structures (LCS), are similar to separatrices thatdivide the flow into dynamically distinct regions. LCS are extensions ofstable and unstable manifolds to general time-dependent flows (see,e.g., G. Haller and G. Yuan, “Lagrangian coherent structures and mixingin two-dimensional turbulence,” Phys. D, vol. 147, pp. 352-370 (December2000)) and they carry a great deal of global information about thedynamics of the flows.

For two-dimensional (2D) flows, LCS are analogous to ridges defined bylocal maximum instability, and can be quantified by local measures ofFinite-Time Lyapunov Exponents (FTLE) (S. C. Shadden, F. Lekien, and J.Marsden, “Definition and properties of Lagrangian coherent structuresfrom finite-time Lyapunov exponents in two-dimensional aperiodic flows,”Physica D: Nonlinear Phenomena, vol. 212, no. 3-4, pp. 271-304 (2005)).Recently, LCS have been shown to coincide with optimal trajectories inthe ocean which minimize the energy and the time needed to traverse fromone point to another (see, e.g., T. Inane, S. Shadden, and J. Marsden,“Optimal trajectory generation in ocean flows,” in Proceedings of the2005 American Control Conference, pp. 674-679, 2005, and C. Senatore andS. Ross, “Fuel-efficient navigation in complex flows,” in Proceedings ofthe 2008 American Control Conference, pp. 1244-1248, 2008). Furthermore,to improve weather and climate forecasting, and to better understandvarious physical, chemical, and geophysical processes in the ocean,there has been significant interest in the deployment of autonomoussensors to measure a variety of quantities of interest. One drawback tooperating sensors in time-dependent and stochastic environments like theocean is that the sensors will tend to escape from their monitoringregion of interest. Since the LCS are inherently unstable and denoteregions of the flow where more escape events may occur (see, e.g., E.Forgoston, L. Billings, P. Yecko, and I. B. Schwartz, “Set-based corralcontrol in stochastic dynamical systems: Making almost invariant setsmore invariant,” Chaos, vol. 21, 013116, 2011), knowledge of the LCS areof paramount importance in maintaining a sensor in a particularmonitoring region.

Existing work in cooperative boundary tracking for robotic teams thatrelies on one-dimensional (1D) parameterizations include C. Hsieh, Z.Jin, D. Marthaler, B. Nguyen, D. Tung, A. Bertozzi, and R. Murray,“Experimental validation of an algorithm for cooperative boundarytracking,” in Proceedings of the 2005 American Control Conference, pp.1078-1083, 2005, S. Susca, S. Martinez, and F. Bullo, “Monitoringenvironmental boundaries with a robotic sensor network,” IEEE Trans. onControl Systems Technology, vol. 16, no. 2, pp. 288-296, 2008, I.Triandaf and I. B. Schwartz, “A collective motion algorithm for trackingtime-dependent boundaries,” Mathematics and Computers in Simulation,vol. 70, pp. 187-202, (2005) and V. M. Goncalves, L. C. A. Pimenta, C.A. Maia, B. Dutra, and G. A. S. Pereira, “Vector fields for robotnavigation along time-varying curves in n-dimensions,” IEEE Trans. onRobotics, vol. 26, no. 4, pp. 647-659 (2010), for static andtime-dependent cases respectively. Formation control strategies fordistributed estimation of level surfaces and scalar fields in the oceanare presented in F. Zhang, D. M. Fratantoni, D. Paley, J. Lund, and N.E. Leonard, “Control of coordinated patterns for ocean sampling,” Int.Journal of Control, vol. 80, no. 7, pp. 1186-1199 (2007), K. M. Lynch,P. Schwartz, I. B. Yang, and R. A. Freeman, “Decentralized environmentalmodeling by mobile sensor networks,” IEEE Trans. on Robotics, vol. 24,no. 3, pp. 710-724 (2008), and W. Wu and F. Zhang, “Cooperativeexploration of level surfaces of three dimensional scalar fields,”Automatica, the IFAC Journal, vol. 47, no. 9, pp. 2044-2051 (2011), andpattern formation for surveillance and monitoring by robot teams isdiscussed in J. Spletzer and R. Fierro, “Optimal positioning strategiesfor shape changes in robot teams,” in Proceedings of the IEEE Int. Conf.on Robotics & Automation, Barcelona, Spain pp. 754-759, 2005, S.Kalantar and U. R. Zimmer, “Distributed shape control of homogeneousswarms of autonomous underwater vehicles,” Autonomous Robots (intl.journal), 2006, and M. A. Hsieh, S. Loizou, and V. Kumar, “Stabilizationof multiple robots on stable orbits via local sensing,” in Proceedingsof the Int. Conf. on Robotics & Automation (ICRA), 2007).

BRIEF SUMMARY OF THE INVENTION

According to the invention, a collaborative control method for trackingLagrangian coherent structures (LCSs) and manifolds on flows employs atleast three autonomous sensors each equipped with a local flow sensorfor sensing flow in a designated fluid medium, e.g. water or air. Afirst flow sensor is a tracking sensor while the other sensors areherding sensors for controlling and determining the actions of thetracking sensor. The tracking sensor is positioned with respect to theherding sensors in the fluid medium such that the herding sensorsmaintain a straddle formation across a boundary; obtaining a local fluidflow velocity measurement from each sensor. A global fluid flow coherentstructure is predicted based on the local flow velocity measurements. Ina water medium, mobile autonomous underwater flow sensors may bedeployed with each tethered to a watersurface craft.

The invention advantageously uses cooperative robots to find coherentstructures without requiring a global picture of the ocean dynamics, andenables a team of robots to track the stable/unstable manifolds ofgeneral 2D conservative flows through local sensing alone. The inventionprovides tracking strategies for mapping LCS in the ocean using AUVs,using nonlinear dynamical and chaotic system analysis techniques tocreate a tracking strategy for a team of robots. The cooperative controlstrategy leverages the spatio-temporal sensing capabilities of a team ofnetworked robots to track the boundaries separating the regions in phasespace that support distinct dynamical behavior. Additionally, boundarytracking relies solely on local measurements of the velocity field. Themethod of the invention may be generally applied to any conservativeflow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows three robots tracking the stable structure B_(S) in aconservative vector field according to the invention;

FIGS. 2A-B show trajectories of 3 robots tracking a sinusoidal boundary(FIG. 2A) and a star-shaped boundary (FIG. 2B) according to theinvention;

FIGS. 3A-B show trajectories of a 3 robot team tracking a star shape(FIG. 3A) and a snapshot of the multi-robot experiment (FIG. 3B)according to the invention;

FIG. 4 shows a phase portrait of a time-independent double-gyre modelaccording to the invention; and

FIGS. 5A-5H show snapshots of the trajectories of the team of 3 robotstracking Lagrangian coherent structures according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

We consider the problem of controlling a team of N planar autonomousunderwater vehicles (“AUVs”) or mobile autonomous underwater flowsensors, e.g. where each underwater flow sensor is tethered to a mobileautonomous watersurface craft, to collaboratively track the materiallines that separate regions of flow with distinct fluid dynamics. Thisis similar to the problem of tracking the stable (and unstable)manifolds of a general nonlinear dynamical system where the manifoldsseparate regions in phase space with distinct dynamical behaviors. Weassume the following 2D kinematic model for each of the AUVs:dx _(i) /dt=V _(i) cos θ_(i) +u _(i,)  (1a)dy _(i) /dt=V _(i) sin θ_(i) +v _(i;)  (1b)where x_(i)=[x_(i),y_(i)]^(T) is the vehicle's planar position, V_(i)and θ_(i) are the vehicle's linear speed and heading, andu_(i)=[u_(i),v_(i)]^(T) is the velocity of the fluid currentexperienced/measured by the i^(th) vehicle. Additionally, we assume eachagent can be circumscribed by a circle of radius r, i.e., each vehiclecan be equivalently described as a disk of radius r.

In this work, u_(i) is provided by a 2D planar conservative vector fielddescribed by a differential equation of the formdx/dt=F(x).  (2)In essence, u_(i)=F_(x)(x_(i)) and v_(i)=F_(y)(x_(i)). Let B_(S) andB_(U) denote the stable and unstable manifolds of Eq. (2). In general,B_(S) and B_(U) are the separating boundaries between regions in phasespace with distinct dynamics. For 2D flows, B_(*) are simplyone-dimensional curves where * denotes either stable (S) or unstable (U)boundaries. For a small region centered about a point on B_(*), thesystem is unstable in one dimension. Finally, let ρ(B_(*)) denote theradius of curvature of B_(*) and assume that the minimum of the radiusof curvature ρ_(min)(B_(*))>r. This last assumption is needed to ensurethe robots do not lose track of the B_(*) due to sharp turns.

The objective is to develop a collaborative strategy to enable a team ofrobots to track B_(*) in general 2D planar conservative flow fieldsthrough local sampling of the velocity field. While the focus is on thedevelopment of a tracking strategy for B_(S), the method can be easilyextended to track B_(U) since B_(U) are simply stable manifolds of Eq.(2) for t<0.

The PIM Triple Procedure

The method of the invention originates from the Proper Interior Maximum(PIM) Triple Procedure, H. E. Nusse and J. A. Yorke, “A procedure forfinding numerical trajectories on chaotic saddles,” Physica D NonlinearPhenomena, vol. 36, pp. 137-156, 1989 (hereinafter “Nusse et al.”)—anumerical technique designed to find stationary trajectories in chaoticregions with no attractors. While the original procedure was developedfor chaotic dynamical systems, the approach can be employed to revealthe stable set of a saddle point of a general nonlinear dynamicalsystem. The procedure consists of iteratively finding an appropriate PIMTriple on a saddle straddling line segment and propagating the tripleforward in time.

Given the dynamical system described by Eq. (2), let DεR² be a closedand bounded set such that D does not contain any attractors of Eq. (2).Given a point xεD, the escape time of x, denoted by T_(E)(x), is thetime x takes to leave the region D under the differential map given byEq. (2).

Let J be a line segment that crosses the stable set B_(S) in D, i.e.,the endpoints of the J are on opposite sides of B_(S). Let{x_(L),x_(C),x_(R)} denote a set of three points in J such that x_(C)denotes the interior point. Then {x_(L),x_(C),x_(R)} is an InteriorMaximum triple if T_(E)(x_(C))>max{T_(E)(x_(L)),T_(E)(x_(R))}.Furthermore, {x_(L),x_(C),x_(R)} is a Proper Interior Maximum (PIM)triple if it is an interior maximum triple and the interval [x_(L),x_(R)] in J is a proper subset off. Then the numerical computation ofany PIM triple can be obtained iteratively starting with an initialsaddle straddle line segment J₀. Let x_(L0) and X_(R0) denote theendpoints of J₀ and apply an ε₀>0 discretization of J₀ such thatx_(L0)=q₀<q₁< . . . <q_(M)=x_(R0). For every point q_(i), determineT_(E)(q_(i)) by propagating q_(i) forward in time using Eq. (2). Thenthe PIM triple in J₀ is given by the points {q_(k−1),q_(k),q_(k+1)}where q_(k)=argmax_(i=1, . . . , M)T_(E)(q_(i)) This PIM triple can thenbe further refined by choosing J₁ to be the line segment containing{q_(k−1),q_(k),q_(k+1)} and reapplying the procedure with another ε₁>0discretization where ε₁<ε₀.

Given an initial saddle straddling line segment J₀, it has been shownthat the line segment given by any subsequent PIM triple on J₀ is also asaddle straddling line segment [H. E. Nusse and J. A. Yorke, “Aprocedure for finding numerical trajectories on chaotic saddles,”Physica D Nonlinear Phenomena, vol. 36, pp. 137-156, 1989.].Furthermore, if we use a PIM triple x(t)={x_(L),x_(C),x_(R)} as theinitial conditions for the dynamical system given by Eq. (2) andpropagate the system forward in time by Δt, then the line segmentcontaining the set x(t+Δt), J_(t+Δt), remains a saddle straddle linesegment. As such, the same numerical procedure can be employed todetermine an appropriate PIM triple on J_(t+Δt). This procedure can berepeated to eventually reveal the entire stable set B_(S) and unstableset B_(U) within D if time was propagated forwards and backwardsrespectively. Furthermore, since the procedure always begins with avalid saddle straddling line segment, by construction, the procedurealways results in a non-empty set.

Building upon the PIM Triple Procedure, as described below the inventionutilizes a cooperative saddle straddle control strategy for a team ofN≧3 robots to track the stable (and unstable) manifolds of a generalconservative time-independent flow field F(x). The invention differsfrom the PIM procedure where it relies solely on information gatheredvia local sensing and shared through the network. In contrast, astraight implementation of the PIM Triple Procedure necessitates globalknowledge of the structure of the system dynamics throughout a givenregion given its reliance on computing escape times.

Controller Synthesis

Consider a team of three robots and identify them as robots {L,C,R}.While the robots may be equipped with similar sensing and actuationcapabilities, we propose a heterogeneous cooperative control strategy.

Let x(0)=[x_(L) ^(T)(0),x_(C) ^(T)(0),x_(R) ^(T)(0)]^(T) be the initialconditions for the three robots. Assume that x(0) lies on the linesegment J₀ where J₀ is a saddle straddle line segment and{x_(L)(0),x_(C)(0),x_(R)(0)} constitutes a PIM triple. Similar to thePIM Triple Procedure, the objective is to enable the robots to maintaina formation such that a valid saddle straddle line segment can bemaintained between robots L and R. Instead of computing the escape timesfor points on J₀ as proposed by the PIM Triple Procedure, robot C mustremain close to B_(S) using only local measurements of the velocityfield provided by the rest of the team. As such, we refer to robot C asthe tracker of the team while robots L and R maintains a straddleformation across the boundary at all times. Robots L and R may bethought of herding robots, since they control and determine the actionsof the tracking robot.

Straddling Formation Control

The controller for the straddling robots consists of two discretestates: a passive control state, U_(P), and an active control state,U_(A). The robots initialize in the passive state U_(P) where theobjective is to follow the flow of the ambient vector field. Therefore,V_(i)=0 for i=L,R. Robots execute U_(P) until they reach the maximumallowable separation distance d_(Max) from robot C. When∥x_(i)-x_(C)∥>d_(max) robot i switches to the active control state,U_(A), where the objective is to navigate to a point p_(i) on thecurrent projected saddle straddle line segment Ĵ_(t) such that,∥p_(i)-p_(C)∥=d_(Min) and p_(C) denotes the midpoint of Ĵ_(t). Whenrobots execute U_(A), V_(i)∥p_(i)-x_(i)-u_(i)∥ and θ_(i)(t)=α_(i)(t)where α_(i) is the angle between the desired, (p_(i)-x_(i)), and currentheading, u_(i), of robot i as shown in FIG. 1. In summary, thestraddling control strategy for robots L and R is given by

$\begin{matrix}{V_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{M\; i\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\{{( {p_{i} - x_{i}} ) - u_{i}}} & {{otherwise},}\end{matrix} } & ( {3a} ) \\{\theta_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{M\; i\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\\alpha_{i} & {{otherwise}.}\end{matrix} } & ( {3b} )\end{matrix}$We note that while the primary control objective for robots L and R isto maintain a straddle formation across B_(S), robots L and R are alsoconstantly sampling the velocity of the local vector field andcommunicating these measurements and their relative positions to robotC. Robot C is then tasked to use these measurements to track theposition of B_(S).

Manifold Tracking Control

Let û_(L)(t), û_(C)(t), and û_(C)(t) denote the current velocitymeasurements obtained by robots L, C, and R at their respectivepositions. Let d(•,•) denote the Euclidean distance function and assumethat d(x_(C),B_(S))<ε such that ε>0 is small. Given the straddle linesegment J_(t) such that x_(L)(k) and x_(R)(k) are the endpoints ofJ_(t), we consider an ε_(t)<ε discretization of J_(t) such thatx_(L)=q₁<q₂< . . . <q_(M)=x_(R). The objective is to use the velocitymeasurements provided by the team to interpolate the vector field at thepoints q₁, . . . , q_(M). Since Eq. (2) has C¹ continuity and if x_(C)is ε-close to B_(S), then the pointq_(B)=argmax_(k=1, . . . , M)u(q_(k))^(T)û_(C)(t) should be δ-close toB_(S) where ε<δ<A and A is a small enough positive constant.

While there are numerous vector field interpolation techniques available(J. C. Agui and J. Jimenez, “On the performance of particle tracking,”Journal of Fluid Mechanics, vol. 185, pp. 447-468, 1987, and E. J.Fuselier and G. B. Wright, “Stability and error estimates for vectorfield interpolation and decomposition on the sphere with rbfs,” SIAM J.Numer. Anal., vol. 47, pp. 3213-3239, 2009), we employ the inversedistance weighting method. For a given set of velocity measurementsû_(i)(t) and corresponding position estimates {circumflex over(x)}_(i)(t), the velocity vector at some point q_(k) is given by

${u( q_{k} )} = {\sum\limits_{j}{\sum\limits_{i = 1}^{N}\frac{w_{ij}{{\hat{u}}_{i}(j)}}{\sum\limits_{j}{\sum\limits_{i = 1}^{N}w_{ij}}}}}$where w_(ij)=∥{circumflex over (x)}_(i)(j)−q_(i)∥⁻². Rather than relysolely on the current measurements provided by the three robots, it ispossible to include the recent history of û_(i)(t) to improve theestimate of u(q_(k)), i.e., û_(i)(t−ΔT), û_(i)(t−2ΔT), and so on, whereΔT is the sampling period and i={L,C,R}. Thus, the control strategy forthe tracking robot C is given byV _(C)=∥[(q _(B) +bû _(B))−x _(C) ]−u _(C)∥  (4a)θ_(C)=β_(C)  (4b)where β_(C) denotes the difference in the heading of robot C and thevector (q_(B)−û_(B)) and b>r is a small number. The term bû_(B) isincluded to ensure that the control strategy aims for a point in frontof robot C rather than behind it. As such, the projected saddle straddleline segment Ĵ_(t) at each time step is given by p_(c)=q_(C)+bu_(C) withĴ_(t) orthogonal to B_(S) at q_(C) and ∥Ĵ_(t)∥ chosen to be in theinterval [2d_(Min)2d_(Max)].

Analysis

Regarding the implementation of the saddle straddle control strategy, webegin with the following key assumption on the robots' initialpositions.

Assumption 1 Given a team of three robots {L,C,R}, assume thatd(x_(C)(0),B_(S))<ε for a small value of ε>0,∥x_(L)-x_(C)∥=|∥x_(R)-x∥=d_(Min) with d_(Min)>2r, and robots L and R areon opposite sides of B_(S).

In other words, assume that the robots initialize in a valid PIM tripleformation and their positions form a saddle straddle line segmentorthogonal to B_(S). Our main result concerns the validity of the saddlestraddle control strategy.

Theorem 1 Given a team of 3 robots with kinematics given by Eq. (1) andu_(i) given by Eq. (2), the feedback control strategy Eq. (3) and Eq.(4) maintains a valid saddle straddle line segment in the time interval[t,t+Δt] if the initial positions of the robots, x(t), is a valid PIMtriple.

The above theorem guarantees that for any given time interval [t,t+Δt]the team maintains a valid PIM triple formation. As such, the iterativeapplication of the proposed control strategy leads to the followingproposition.

Proposition 1 Given a team of 3 robots with kinematics given by Eq. (1)and u_(i) given by Eq. (2), the feedback control strategy results in anestimate of B_(S) denoted as {circumflex over (B)}_(S), such that<B_(S), {circumflex over (B)}_(S)>_(L2)<W for some W>0 where <•,•>_(L2)denotes the inner product (which provides an L₂ measure between theB_(S) and {circumflex over (B)}_(S) curves).

From Theorem 1, since the team is able to maintain a valid PIM tripleformation across B_(S) for any given time interval [t,t+Δt], thisensures that an estimate of B_(S) in the given time interval alsoexists. Applying this reasoning in a recursive fashion, one can showthat an estimate of B_(S) can be obtained for any arbitrary timeinterval. Preferably, one also determines the bound on W such that{circumflex over (B)}_(S) results in a good enough approximation since Wdepends on the sensor and actuation noise, the vector interpolationroutine, the sampling frequency, and the time scales of the flowdynamics.

Results

Simulations:

We illustrate the proposed control strategy given by Eq. (3) and Eq. (4)with the following simulation results. FIG. 2A shows the trajectories ofthree robots tracking a sinusoidal boundary while FIG. 2B shows the teamtracking a 1D star-shaped boundary. We note that throughout the entirelength of the simulation, the team maintains a saddle straddle formationacross the boundary.

In both examples, u=−a∇φ−b∇×ψ where a,b>0 and φ(x) is an artificialpotential function such that φ(x)=0 for all xεB_(*) and φ(x)<0 for anyxεR²/B_(*). The vector ψ is a 3×1 vector whose entries are given by[0,0,γ(x,y)]^(T) where γ(x,y) is the curve describing the desiredboundary. Lastly, the estimated position of the boundary is given by theposition of the tracking robot, i.e., robot C. In these examples, wefiltered the boundary position using a simple first-order low passfilter.

Experiments

We also implemented the control strategy on our multi-robot testbed. Thetestbed consisted of three mSRV-1 robots in a 4.8×5.4 meter workspace.The mSRV-1 are differential-drive robots equipped with an embeddedprocessor, color camera, and 802.11 wireless capability. Localizationfor each robot was provided via a network of overhead cameras. FIG. 3Ashows the trajectories of the robots tracking a star shaped boundary.FIG. 3B is a snapshot of the experimental run.

Extension to Periodic Boundaries

Next, we consider the system of 3 robots with kinematics given by Eq.(1) where u_(i) is determined by the wind-driven double-gyre flow modelwith noise

$\begin{matrix}{\overset{.}{x} = {{{- \pi}\; A\;{\sin( {\pi\;\frac{f( {x,t} )}{s}} )}{\cos( {\pi\;\frac{y}{s}} )}} - {\mu\; x} + {{\eta_{1}(t)}.}}} & ( {5a} ) \\{{\overset{.}{y} = {{\pi\; A\;{\cos( {\pi\;\frac{f( {x,t} )}{s}} )}{\sin( {\pi\;\frac{y}{s}} )}\frac{\mathbb{d}f}{\mathbb{d}x}} - {\mu\; y} + {\eta_{2}(t)}}},} & ( {5b} ) \\{{f( {x,t} )} = {{{{ɛsin}( {{\omega\; t} + \psi} )}x^{2}} + {( {1 - {2ɛ\;{\sin( {{\omega\; t} + \psi} )}}} ){x.}}}} & ( {5c} )\end{matrix}$

When ε=0, the double-gyre flow is time-independent, while for ε≠0, thegyres undergo a periodic expansion and contraction in the x direction.In Eq. (5a-c), A approximately determines the amplitude of the velocityvectors, ω/2π gives the oscillation frequency, ε determines theamplitude of the left-right motion of the separatrix between the gyres,ψ is the phase, μ determines the dissipation, s scales the dimensions ofthe workspace, and η_(i)(t) describes a stochastic white noise with meanzero and standard deviation σ=√{square root over (2I)}, for noiseintensity I. In this work, η_(i)(t) can be viewed as either measurementor environmental noise. FIG. 4 shows the phase portrait of thetime-independent double-gyre model.

FIGS. 5A-5H show trajectories of the team of 3 robots trackingLagrangian coherent structures of the system described by Eq. (5a-c)with A=10, μ=0.005, ε=0.1, ψ=0, I=0.01, and s=50. The trajectories ofthe straddling robots are shown in black and the estimated LCS is shownin white.

While the present invention has been described with respect to exemplaryembodiments thereof, it will be understood by those of ordinary skill inthe art that variations and modifications can be effected within thescope and spirit of the invention. For example, the invention may beapplied to any designated fluid medium where sensors may communicate andtake local position measurements, including but not limited toautonomous air vehicle sensors operating in air where sensors maycommunicate and take local position measurements. Also, instead ofemploying AUVs, mobile autonomous underwater flow sensors may bedeployed with each sensor tethered to an autonomous watersurface craft.

The invention claimed is:
 1. A collaborative control method for trackingLagrangian coherent structures (LCSs) and manifolds on flows using atleast three autonomous sensors each equipped with a local flow sensorfor sensing flow in an air medium, and wherein a first flow sensor is atracking sensor and the other sensors are herding sensors forcontrolling and determining the actions of the tracking sensor,comprising: deploying the sensors in the air medium whereby the trackingsensor is positioned with respect to the herding sensors such that theherding sensors maintain a straddle formation across a boundary;obtaining a local fluid flow velocity measurement from each sensor; andbased on the local flow velocity measurements predicting a global fluidflow coherent structure.
 2. The method of claim 1, wherein a straddlingcontrol strategy is given by $\begin{matrix}{V_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{{Mi}\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\{{( {p_{i} - x_{i}} ) - u_{i}}} & {{otherwise},}\end{matrix} } & ( {3a} ) \\{\theta_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{M\; i\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\\alpha_{i} & {{otherwise}.}\end{matrix} } & ( {3b} )\end{matrix}$ where d_(Max) is a maximum allowable separation distanceof a pair of straddling robots L and R from a control robot C., anobjective is to navigate to a point p_(i) on a current projected saddlestraddle line segment Ĵ_(t) such that, ∥p_(i)−p_(C)∥=d_(Min) and p_(C)denotes the midpoint of Ĵ_(t), and α_(i) is an angle between a desired,(p_(i)−x_(i)), and a current heading u_(i) of a robot i.
 3. Acollaborative control method for tracking Lagrangian coherent structures(LCSs) and manifolds on flows using at least three mobile autonomousunderwater flow sensors, and wherein a first flow sensor is a trackingsensor and the other sensors are herding sensors for controlling anddetermining the actions of the tracking sensor, and wherein the sensorsare each tethered to an autonomous watersurface craft, comprising:deploying the sensors in a body of water whereby the tracking sensor ispositioned with respect to the herding sensors such that the herdingsensors maintain a straddle formation across a boundary; obtaining alocal flow velocity measurement from each sensor; and based on the localflow velocity measurements predicting a global flow structure useful forplotting an optimal course for a vessel between two or more locations.4. The method of claim 3, wherein a straddling control strategy is givenby $\begin{matrix}{V_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{M\; i\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\{{( {p_{i} - x_{i}} ) - u_{i}}} & {{otherwise},}\end{matrix} } & ( {3a} ) \\{\theta_{i} = \{ \begin{matrix}0 & {{{if}\mspace{14mu} d_{M\; i\; n}} < {{x_{i} - x_{C}}} < d_{{Ma}\; x}} \\\alpha_{i} & {{otherwise}.}\end{matrix} } & ( {3b} )\end{matrix}$ where d_(Max) is a maximum allowable separation distanceof a pair of straddling robots L and R from a control robot C., anobjective is to navigate to a point p_(i) on a current projected saddlestraddle line segment Ĵ_(t) such that, ∥p_(i)−p_(C)∥=d_(Min) and p_(C)denotes the midpoint of Ĵ_(t), and α_(i) is an angle between a desired,(p_(i)−x_(i)), and a current heading u_(i) of a robot i.